Problem: Solve the exponential equation for $x$. 125 − x − 7 ⋅ 5 6 x − 12 = 125 2 x + 3 125\^{-x-7}\cdot 5\^{ 6x-12}=125\^{ 2x+3} $x=$
Solution: The strategy Let's write $5$ in base $125$. Then, using the properties of exponents, we can express the entire left hand side of the equation as $125$ raised to some linear function. Finally, we can equate the exponents of the resulting equation to solve for the unknown. Simplifying the left hand side 125 − x − 7 ⋅ 5 6 x − 12 = 125 − x − 7 ⋅ ( 125 1 3 ) 6 x − 12 = 125 − x − 7 ⋅ 125 2 x − 4 = 125 − x − 7 + ( 2 x − 4 ) = 125 x − 11 ( 5 = 125 1 3 ) ( ( a n ) m = a n ⋅ m ) ( a n ⋅ a m = a n + m ) \begin{aligned} 125\^{-x-7}\cdot 5\^{ 6x-12}&=125\^{-x-7}\cdot (125\^{ \frac13})\^{ 6x-12} &&&&(5=125\^{ \frac13})\\\\ &=125\^{C{-x-7}}\cdot 125\^{ {2x-4}}&&&&((a^n)^m=a^{n\cdot m}) \\\\ &=125\^{ C{-x-7} \ + \ ({2x-4}) }&&&&(a^n\cdot a^m=a^{n + \normalsize m})\\\\ &=125\^{ x-11} \end{aligned} Solving the linear equation We obtain the following equation. 125 x − 11 = 125 2 x + 3 125\^{ x-11}=125\^{ 2x+3} Now we can equate the exponents and solve for $x$. $\begin{aligned} x-11 &=2x+3\\\\ x &= -14\end{aligned}$ The answer The answer is $x=-14$. You can check this answer by substituting $\it{x=-14}$ in the original equation and evaluating both sides.